alcs lect03fv

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Advanced Linear Control

Systems

Dr. Zeeshan Khan

Department of Electrical Engineering

Center for Emerging Sciences, Engineering & Technology (CESET), Islamabad

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Course Objectives

The course addresses dynamic systems, i.e., systems that

evolve with time.

Systems that can be modeled by Ordinary Differential

Equations (ODEs), and that satisfy certain linearity and time-

invariance conditions.

Special consideration on MIMO systems.

We will analyze the response of these systems to inputs and

initial conditions: for example, stability and performance

issues will be addressed. It is of particular interest to analyze

systems obtained as interconnections (e.g., feedback) of two

or more other systems.

We will learn how to design (control) systems that ensure

desirable properties (e.g., stability, performance) of the

interconnection with a given dynamic system. 2


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Course Outline

The course will be structured in several major sections:

A review of linear algebra, and of least squares problems.

Representation, structure, and behavior of multi-input,multi-output (MIMO) linear time-invariant (LTI) systems.

Robust Stability and Performance. Approaches to optimaland robust control design.

Hopefully, the material learned in this course will form a

valuable foundation for further work in systems, control,estimation, identification, signal processing, andcommunications.

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Assignments, Quizzes and Exams

At least 3 Assignments will be given during the course

At least 6 quizzes will be taken

There will be 2 Exams

o 1 Mid-termo 1 Final exam

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Grading Policy

10%

10%

30%

50%Assignments

Quizzes

Mid term

Final

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Notes and Texts

Course Book:

Stanislaw H. Zak, Systems and Control, Oxford UniversityPress 2003.

For beginners in Control:

Ogata, K. Modern Control Engineering, Prentice Hall.

Linear Algebra, Schaums Series.

More references:

D.G. Luenberger, Introduction to Dynamic Systems, Wiley,1979.

T. Kailath, Linear Systems, Prentice-Hall, 1980.

J.C. Doyle, B.A. Francis, and A.R. Tannenbaum, FeedbackControl Theory, Macmillan, 1992.

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Tentative Schedule

Lecture # Date Topic Chapter

1 02-03-2013 Introduction to dynamic systems and control,

Matrix algebra

Appendix

2 03-03-2013 Projection theorem, Least squares estimation Appendix

3 09-03-2013 Dynamical Systems and Modeling Chapter 1

4 10-03-2013 Mathematical Modeling and Examples Chapter 1

5 16-03-2013 Analysis of Modeling Equations Chapter 2

6 17-03-2013 Linearization differential equations Chapter 2

7 23-03-2013 Describing Functions for Nonlinear Systems Chapter 2

8 24-03-2013 Reachability, Controllability, Observability etc Chapter 3

9 30-03-2013 Companion forms and linear state feedback Chapter 3

10 31-03-2013 State Estimator and combined Controller Estimator Chapter 3

11 06-04-2013 Stability and methods to determine stability Chapter 4

12 07-04-2013 Stability of nonlinear systems and Lyapunov

theorems

Chapter 4

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What Is a System?

A system is characterized by two properties, which are

as follows:

1. The interrelations between the components that are

contained within the system

2. The system boundaries that separate the components

within the system from the components outside

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What is a dynamic system?

A dynamical system consists of a set of

possible states, together with a rule that

determines the present state in terms of past

states.

The system quantities whose behavior can be

measured or observed are referred to as the

system outputs.

SystemInput output

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Describing a Control Problem

The essential elements of the control problem, as

described by Owens are as follows:

1. A specified objective for the system

2. A model of a dynamical system to be controlled

3. A set of admissible controllers

4. A means of measuring the performance of any

given control strategy to evaluate its

effectiveness

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Modeling a Dynamic System

A common model of a dynamical system is the finite set of ordinary

differential equations of the form

x(t) = f (t, x(t), u(t)), x(t0) = x0,

y(t) = h(t, x(t), u(t)),

Where, the statexRn, the input u Rm, the output yRp, and f and h

are vector-valuedfunctions withf : RRn Rm Rn and h : RRn Rm Rp.

Another common modelof a dynamical system is the finite set of

difference equations,

x(k + 1) = f (k, x(k), u(k)), x(k0) = x0,

y(k) = h(k, x(k), u(k)),

Where,x(k) = x(kh), u(k) = u(kh), h is the sampling interval, and k 0 is an

integer.

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Open-Loop Versus Closed-Loop

We distinguish between two types of control

systems. They are:

Open-loop control systems

Closed-loop control systems

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Open loop System

An open-loop control system usually contains thefollowing:

1. A process to be controlled, labeledplant

2. The controlling variable of the plant, called theplantinput, or justinputfor short

3. The controlled variable of the plant, called theplantoutput, or justoutputfor short

4. A reference input, which dictates the desired value ofthe output

5. A controllerthat acts upon the reference input in order

to form the system input forcing the output behaviorin accordance with the reference signal

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Connecting Feedback and the

Summing Point

6. Thefeedback loop where the output signal is measured

with a sensor and then the measuredsignal is fed back

to the summing junction

7. The summing junction, where the measured outputsignal is subtracted from the reference (command)

input signal in order to generate an error signal, also

labeled as an actuating signal

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Closed Loop System

In a closed-loop system the error signal causes anappropriate action of the controller, which in turninstructs the plant to behave in a certain way in orderto approach the desired output, as specified by thereference input signal.

Thus, in the closed-loop system, the plant outputinformation is fed back to the controller, and thecontroller then appropriately modifies the plant outputbehavior.

A controller, also called a compensator, can be placedeither in the forward loop, as in Figure, or in thefeedback loop.

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Closed Loop System (Contd )

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Axiomatic Definition of a Dynamical

System Following Kalman, we can define a dynamical system

formally using the following axioms:

1. There is given a state spaceX; there is also T , aninterval in the real line representing time.

2. There is given a space U of functions on T thatrepresent the inputs to the system.

3. For any initial time t0 in T , any initial statex0in X, andany input u in U defined fort t0 the future states of

the system are determined by the transition mapping

: T X U Xwritten as(t1; t0,x(t0), u(t)) = x(t1).

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System (Contd )4. The identity property of the transition mapping, that is,

(t0; t0,x(t0), u(t0)) = x(t0).

5. The semigroup property of the transition mapping, that is,

(t2; t0,x(t0), u(t)) = (t2; t1,(t1; t0, x(t0), u(t)), u(t)).

The semigroup property axiom states that it is irrelevant whetherthe system arrives at the state at time t2 by a direct transition fromthe state at time t0, or by first going to an intermediate state attime t1, and then having been restarted from the state at time t1andmoving to the state at time t2. In either case the system,satisfying the semigroup axiom, will arrive at the same state at timet2. This property is illustrated in Figure.

c

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System (Contd )6. The causality property, that is,(t; t0,x(t0), u1(t)) =(t; t0, x(t0),

u2(t)) for t0, tTif and only ifu1(t) = u2(t) for all tT .

7. Every output of the system is a function of the form

h : T X U Y,

where Y is the output space.

8. The transition mapping and the output mapping h arecontinuous functions.

9. If in addition the system satisfies the following axiom, then it is saidto be time-invariant:

(t1; t0,x(t0), u(t)) = (t1 + ; t0+ , x(t0), u(t)),

where t0, t1 T .

Thus, a dynamical system can be defined formally as a quintuple{T,X,U,Y} satisfying the above axioms.

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System (Contd ) Our attention will be focused on dynamical systems modeled by a set of

ordinary differential equations xi = fi (t, x1, x2, . . . , xn, u1, u2, . . . , um), xi

(t0) = xi0, i = 1, 2, . . . , n, together withp functions, yj= h j (t, x1, x2, . . . ,

xn, u1, u2, . . . , um), j = 1, 2, . . . , p.

The system model state isx = [x1 x2 xn]T

Rn. The system input is u =[u1 u2 um]TRm, and the system output is y = [y1 y2 yp ]TRp.

In vector notation the above system model has the form

x = f (t, x, u), x(t0) = x0,

y = h(t, x, u),

wheref : R Rn Rm Rn and h : R Rn Rm Rp are vector-valued

functions.

In further considerations, we regard a dynamical system under

considerations and its model represented by the above equations as

equivalent. We now illustrate dynamical system axioms on a simple

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Mathematical Modeling Process

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Review of Work and Energy Concepts

Suppose we are given a particle of a constant

mass m subjected to a force F. Then, by

Newtons second law we have,

Here the force F and the velocity v are

vectors. Therefore, they can be represented

as

Using the above notation, we represent

equation

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Review (contd )

Suppose now that a force

F is acting on a particle

located at a point A and

the particle moves to a

point B. The work Wdone byF along

infinitesimally smalldistance s is

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Review (contd )

where s = [x1 x2 x3]T. The work WAB

done on the path from A to B is obtained

byintegrating the above equation:

We now would like to establish a relation

between work and kinetic energy. For this

observe that

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Review (contd )

Using the above relation and Newtons second

law, we express WAB in a different way as

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Review (contd )

The above relation can be used to define the

kinetic energy of a particle as the work

required to change its velocity from some

value vA to a final value vB.

The relation WAB = KBKA = Kis also known as

the workenergy theorem for a particle.

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Potential and Kinetic Energy

A force is conservative if the work done by the force on aparticle that moves through any round trip is zero. In otherwords, a force is conservative if the work done by it on aparticle that moves between two points depends only onthese points and not on the path followed.

We now review the notion ofpotential energy, or theenergy of configuration. Recall that ifthe kinetic energy Kof a particle changes by K, then the potential energy U mustchange byan equal but opposite amount so that the sum ofthe two changes is zero; that is, K+U = 0

This is equivalent to saying that any change in the kineticenergy of the particle is compensated for by an equal butopposite change in the potential energy U of the particle sothat their sum remains constant; that is, K + U = constant

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The potential energy of a particle represents a form ofstored energy that can be recovered and convertedinto the kinetic energy.

If we now use the workenergy theorem, then we

obtainW = K = U.

The work done by a conservative force depends onlyon the starting and the end points of motion and noton the path followed between them. Therefore, formotion in one dimension, we obtain

thus,

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Thus, we can write:

A generalized equation to motion in three dimensions yields:

Thus,

There exist a property of a conservative vector field that the workdone by it on a particle that moves between two points dependsonly on these points and not on the path followed.

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We now represent Newtons equation in an

equivalent format that establishes a

connection with the Lagrange equations of

motion, which are discussed in the following

section. To proceed, note that

Hence:

Or:

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Let L = K U;

The function L defined above is called the Lagrangianfunction or just the Lagrangian. Note that

It can be simplified into:

Equations above are called the Lagrange equations ofmotion, in Cartesian coordinates, for a single particle. Theyare just an equivalent representation of Newtonsequations as described above.

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